s_curl(3) curl-like operator for the Stokes stream function computation

## SYNOPSIS

form(const space M, const space& V, "s_curl");

## DESCRIPTION

Assembly the form associated to the s_curl operator on a finite element space V:

```                /
|
b(xi,u) = |  u.s_curl(xi) dx
|
/ Omega

```
The M and V space may be a either P1 or P2 finite element space. The M space is scalar-valued while the V is vector-valued. See also form(2) and space(2).

For cartesian coordinate systems, this form coincide with the usual "curl" one (see curl(3)). In the axisymmetric case:

```                /
| (d xi      d xi   )
b(xi,u) = | (---- ur - ---- uz) r dr dz
| (d z       d r    )
/ Omega

```
The b form is denoted as "s_curl", for Stokes stream function computation (see s_grad_grad(3)) as it is closely related to the "curl" operator (see curl(3)), but differs by the r and 1/r factors, as:

```                   (       d (r xi)     d xi )
curl(xi) = ( (1/r) -------- ; - -----)
(         d r        d z  )

```
while

```                   ( d xi       d xi )
s_curl(xi) = ( ----  ;  - ---- )
( d r        d z  )

```
Notice also that the differentiation is performed on the xi variable here: b(xi,u)=(s_curl(xi),u) while the "curl" form brings the differentiation on the u vector-valued variable: (curl(u),xi), i.e. a transpose formulation.

## ORIENTATION AND SIGN FIX

The (r,theta,z) coordinate system has positive orientation, thus (z,r,theta) and (z,r) are positive also. But (r,z,theta) and (r,z) are negative : the sign of s_curl is then inverted to obtain the same result as if (z,r) was used.

## EXAMPLE

The following piece of code build the form associated to the P1 approximation:

```        geo g("square");
space M(g, "P1");
space V(g, "P1", "vector");
form a(M, V, "s_curl");

```